All designs including electrical circuit designs can be described by operational characteristics called performance functions or PFs. When a design is made into a hardware, it acquires some variability based on the manufacturing parameter fluctuations. The design is considered good when the design functional value or the characteristics varies, within a set limit under these manufacturing fluctuations. Hence circuits are designed for manufacturing fluctuations of the parameters. When these manufacturing parameters are within an accepted range, the design must function according to specifications. Design specifications are sometimes called design objectives. While these manufacturing parameters are generally independent of each other, the circuit PF is usually dependent upon all of them. The variability of a parameter or PF can be specified by a statistical quantity sigma, which is the standard deviation of all values from the median. For example, referring to FIG. 1A, a hypothetical distribution of a variable, which in the present case can be a physical parameter or a circuit performance function, has been shown with a single peak or maximum. The extreme value region of the distribution is called the tail. FIG. 1B shows a magnification of the tail region of FIG. 1A. The subscript E denotes an estimated value, while T denotes a true value (may be obtained by simulation). For the case of a normal distribution, shown in FIG. 1A, the variable is distributed in a bell-shaped curve around its mean value. In this case, a specification of mean .+-.3 sigma means that 99.87 percent of all the circuits (designs) built will have the PF within the specified range. Conversely the number of circuits that are outside the specification will be 0.0013 cases or approximately 1 in a thousand. To accurately determine the tail portion outside of the mean + or -3 sigma, requires determining 1000 cases, so that at least one PF of the distribution will be in the tail region. This is referred to as worst-case analysis.
An electrical circuit used in very large scale integration application such as computers, has several important performance functions that are of interest to the designer. Some of these functions are circuit speed or signal delay, signal gain, noise etc. The circuit is usually defined by the values of its constituent devices, such as resistances, capacitances, inductances, power supply currents, voltages, transistor gains etc. which determine PFs. It is these elements that results from the manufacturing parameters. For example, the value of a silicon resistor is determined by its physical dimensions and the electrical sheet resistivity of the silicon. The sheet resistivity in turn is affected by the concentration and distribution of dopants (impurities) added to the silicon. Variations in the physical dimensions or the dopant concentration lead to a variation of the resistor value. Similarly, a Field Effect Transistor (FET), has parameters such as channel length, channel width, gate oxide thickness, mobility, and effective charge, etc. These parameters determine the FETs functional characteristics. Thus it is seen that the variability in circuit performance as described by its PF can be derived from the variabilities of several parameters.
In order to construct the variations of a performance function, the circuit needs to be analyzed at various combinations of parameter values. Depending on the number of parameters of the circuit, the complexity of the circuit, simulating a circuit for analysis takes up large computing time and hence is expensive. Some of the well known circuit simulation programs are Advanced Statistical and Transient Analysis Program (ASTAP), Circuit Simulator for IC Circuits (SPICE), and Statistical Simulator for IC Fabrication (FABRICS II). These are used to simulate the circuits, devices and processes. Depending on the complexity of the device or circuit, these simulations require large computers and use significant CPU times.
Another aspect of determining the distribution curve is the selection of the parameters for each simulation. In the Monte-Carlo technique, the calculation values used in simulation are totally random combinations of parameter values. Other techniques use different ways to select the combination of parameter values for simulation. Nassif, et al. compares the worst case analysis results using Monte-Carlo techniques with results from use of process disturbances in conjunction with the Fabrics-II simulator (S.R. Nassif, et al., "Fabrics II - A Statistical Simulator of the IC Fabrication Process", Proceedings of International Conference on Circuits and Computers, IEEE, New York, September 1982). F. Severson and S. Simpkins, Custom Integrated Circuit Conference, pp. 114-8, IEEE 1987, discloses a technique that uses Hadamard matrix analysis in conjunction with SPICE to determine worst case values. These techniques use circuit simulation and only differ in how the parameters are picked for analysis. Thus these prior art techniques are extremely expensive for highly complex designs. Although, they accurately determine the entire distribution curve using circuit simulation, the designer is really interested in the tail values. To this extent, a large fraction of their computing effort is wasted as the tail values usually represent only 1 percent or less of the total distribution.